Problem 19 - The Calculus with Analytic Geometry

Question

In Exercises 19 through 22 , use Theorem \(7.6 .1\) to find the indicated derivative. \(D_{x} \int_{0}^{x} \sqrt{4+t^{5}} d t\)

Step-by-Step Solution

Answer

The derivative is \( \sqrt{4+x^{5}} \).

Step 1: Identify the Components

Identify the components of the problem. We need to find the derivative of the integral \( \int_{0}^{x} \sqrt{4+t^{5}} \, dt \) with respect to \( x \).

Step 2: Apply the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, if \( F(x) = \int_{a}^{x} f(t) \, dt \), then the derivative of \( F(x) \) with respect to \( x \) is \( F'(x) = f(x) \).

Step 3: Apply Theorem 7.6.1

Apply theorem (Fundamental Theorem of Calculus) directly: \( D_{x} \int_{0}^{x} \sqrt{4+t^{5}} \, dt = \sqrt{4+x^{5}} \).

Key Concepts

Derivative of an Integral

The concept of the derivative of an integral is rooted in the Fundamental Theorem of Calculus. This theorem forms a bridge between differentiation and integration. Essentially, it tells us that if we have a function that is the integral of another function, the derivative of this integral brings us back to the original function.

In mathematical terms, if you have a function expressed as an integral \(F(x) = \int_{a}^{x} f(t) \, dt\), then the derivative of \(F(x)\) with respect to \(x\) is simply the original function evaluated at \(x\), i.e., \(F'(x) = f(x)\).

So, for the integral \(\int_{0}^{x} \sqrt{4+t^{5}} \, dt\), we apply this concept to find the derivative with respect to \(x\). We end up with \(\sqrt{4+x^{5}}\) because the function inside the integral, \(\sqrt{4+t^{5}}\), is evaluated at \(x\).

Theorem 7.6.1

Theorem 7.6.1 is a specific instance of the Fundamental Theorem of Calculus, focusing on the relationship between differentiation and integration. It emphasizes that the process of taking a derivative and then integrating brings you back to your original function, essentially reversing each other's effects.

To put it in simple terms, this theorem states that if you have a definite integral function \(F(x) = \int_{a}^{x} f(t) \, dt\), then the derivative of this function, \(F'(x)\), is simply the integrand function \(f(x)\) evaluated at \(x\). Therefore, when solving the problem \(D_{x} \int_{0}^{x} \sqrt{4+t^{5}} \, dt\), applying Theorem 7.6.1 directly lets us find that the derivative is \(\sqrt{4+x^{5}}\). This is because you’re taking the derivative of the integral function, which as per the theorem, results in the integrand function evaluated at the upper limit of the integral.

Step-by-step Solution

For clarity and deep understanding, let's break down the solution into steps:

• Step 1: Identify the Components - In the given exercise, we’re being asked to find the derivative of the integral \(D_{x} \int_{0}^{x} \sqrt{4+t^{5}} \, dt\). Note that this problem can be tackled using the Fundamental Theorem of Calculus.


• Step 2: Apply the Fundamental Theorem of Calculus - According to the theorem, if we have an integral of the form \(\int_{a}^{x} f(t) \, dt\), its derivative will be the function inside the integral evaluated at the upper limit of the integral. This is the core insight from the theorem that provides a straightforward solution.


• Step 3: Apply Theorem 7.6.1 - Using Theorem 7.6.1, we know that the derivative of the integral will be \(f(x)\). From our problem, \(f(t)\) is \(\sqrt{4+t^{5}}\), thus the derivative will be \( \sqrt{4+x^{5}} \).

Breaking down each part of the theorem and its application helps demystify the process. This step-by-step approach is especially useful for visual learners who benefit from seeing concepts applied in a structured manner.